This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

 BetaDistribution represents a continuous beta distribution with shape parameters and .
• The probability density for value in a beta distribution is proportional to for , and is zero for or . »
Probability density function:
Cumulative distribution function:
Mean and variance:
Median:
Probability density function:
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Cumulative distribution function:
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Mean and variance:
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Median:
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 Scope   (7)
Generate a set of pseudorandom numbers that are beta distributed:
Compare the histogram to the PDF:
Distribution parameters estimation:
Estimate the distribution parameters from sample data:
Compare a density histogram of the sample with the PDF of the estimated distribution:
Skewness varies with shape parameters:
When both parameters go to , the distribution becomes symmetric:
Kurtosis varies with shape parameters:
In the limit the kurtosis becomes the same as for NormalDistribution:
Different moments with closed forms as functions of parameters:
Hazard function:
Quantile function:
 Applications   (2)
Cloud duration approximately follows a beta distribution with parameters 0.3 and 0.4 for a particular location. Find the probability that cloud duration will be longer than half a day:
Simulate the fraction of the day that is cloudy over a period of one month:
Find the average cloudiness duration for a day:
Find the probability of having exactly 20 days in a month with cloud duration less than 10%:
Find the probability of at least 20 days in a month with cloud duration less than 10%:
Beta distribution can be used to model the proportion of the stocks that increase in value on a given day. Fit beta distribution into Dow Jones Industrial stocks:
Find daily change:
Filter out missing data and pad with zeros:
Calculate the daily ratio of companies with an increase in value:
Find fit, excluding days with zero companies having an increase in value:
Compare the histogram of the data with the PDF of the estimated distribution:
Find the probability that at least 60% of Dow Jones Industrial stocks will increase in value:
Find the average percentage of Dow Jones Industrial stocks that will increase in value:
Simulate the percentage of Dow Jones Industrial stocks that will increase in value for 30 days:
Parameter influence on the CDF for each :
If a variate follows beta distribution, then follows the reflected distribution:
Relationships to other distributions:
BetaPrimeDistribution can be obtained as a transformation of the beta-distributed variable:
Beta distribution is a special case of PearsonDistribution of type 1:
Beta distribution can be obtained as a transformation of GammaDistribution:
Beta distribution can be obtained as a transformation of ChiSquareDistribution:
FRatioDistribution can be obtained from beta distribution:
Beta distribution is an order distribution of variables from UniformDistribution:
ExponentialDistribution is a limit of a scaled beta distribution:
KumaraswamyDistribution is a transformation of beta distribution:
KumaraswamyDistribution simplifies to a special case of beta distribution:
PERTDistribution is a transformation of beta distribution:
Univariate marginals of DirichletDistribution have beta distribution:
BetaDistribution is not defined when either or is not a positive real number:
Substitution of invalid parameters into symbolic outputs gives results that are not meaningful:
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